VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 16.5s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1027519946.5492055

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{4}\\ t_1 := t\_0 \cdot f\\ t_2 := e^{t\_1}\\ t_3 := e^{-t\_1}\\ -\frac{1}{t\_0} \cdot \log \left(\frac{t\_2 + t\_3}{t\_2 - t\_3}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)) (t_1 (* t_0 f)) (t_2 (exp t_1)) (t_3 (exp (- t_1))))
   (- (* (/ 1.0 t_0) (log (/ (+ t_2 t_3) (- t_2 t_3)))))))

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{4}\\ t_1 := t\_0 \cdot f\\ t_2 := e^{t\_1}\\ t_3 := e^{-t\_1}\\ -\frac{1}{t\_0} \cdot \log \left(\frac{t\_2 + t\_3}{t\_2 - t\_3}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)) (t_1 (* t_0 f)) (t_2 (exp t_1)) (t_3 (exp (- t_1))))
   (- (* (/ 1.0 t_0) (log (/ (+ t_2 t_3) (- t_2 t_3)))))))

Alternative 1: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.25\\ \frac{\log \tanh \left(t\_0 \cdot f\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (PI) 0.25))) (/ (log (tanh (* t_0 f))) t_0)))

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)}} \]

5. Final simplification 98.7%

   \[\leadsto \frac{\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right)}{\mathsf{PI}\left(\right) \cdot 0.25} \]
6. Add Preprocessing

Alternative 2: 96.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \mathsf{PI}\left(\right), -0.010416666666666666, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log
   (/
    (fma
     (* (fma (* 4.0 (PI)) -0.010416666666666666 (* 0.125 (PI))) f)
     f
     (/ 4.0 (PI)))
    f))))

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing

3. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

5. Applied rewrites 96.1%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 2\right) \cdot 0.005208333333333333, \left(0.0625 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 96.1%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \mathsf{PI}\left(\right), -0.010416666666666666, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

8. Final simplification 96.1%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \mathsf{PI}\left(\right), -0.010416666666666666, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]
9. Add Preprocessing

Alternative 3: 95.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)}{-0.25 \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (log (/ (/ 4.0 (PI)) f)) (* -0.25 (PI))))

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing

3. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}\right)} \]
4. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

   2. metadata-eval N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{\color{blue}{2 \cdot 1}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right) \]

   3. associate-*r/ N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}}{f}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\frac{2 \cdot 1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}}{f}\right) \]

   6. metadata-eval N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{\color{blue}{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right) \]

   7. distribute-rgt-out-- N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{2}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)}}}{f}\right) \]

   8. metadata-eval N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{2}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}}}{f}\right) \]

   9. *-commutative N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{2}{\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}}}{f}\right) \]

   10. associate-/r* N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\frac{\frac{2}{\frac{1}{2}}}{\mathsf{PI}\left(\right)}}}{f}\right) \]

   11. metadata-eval N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{\color{blue}{4}}{\mathsf{PI}\left(\right)}}{f}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}}}{f}\right) \]

   13. lower-PI.f64 95.5

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{4}{\color{blue}{\mathsf{PI}\left(\right)}}}{f}\right) \]

5. Applied rewrites 95.5%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto -\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   2. div-inv N/A

      \[\leadsto -\frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   3. metadata-eval N/A

      \[\leadsto -\frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   4. *-commutative N/A

      \[\leadsto -\frac{1}{\color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto -\frac{1}{\frac{1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   6. add-sqr-sqrt N/A

      \[\leadsto -\frac{1}{\frac{1}{4} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   7. associate-*r* N/A

      \[\leadsto -\frac{1}{\color{blue}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto -\frac{1}{\color{blue}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto -\frac{1}{\color{blue}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto -\frac{1}{\left(\frac{1}{4} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto -\frac{1}{\left(\frac{1}{4} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto -\frac{1}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   13. lower-sqrt.f64 94.2

       \[\leadsto -\frac{1}{\left(0.25 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

7. Applied rewrites 94.2%

   \[\leadsto -\frac{1}{\color{blue}{\left(0.25 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]
8. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \frac{1}{\left(\frac{1}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

9. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)}{-0.25 \cdot \mathsf{PI}\left(\right)}} \]
10. Add Preprocessing

Alternative 4: 95.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}{\mathsf{PI}\left(\right) \cdot 0.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (log (* (* (PI) f) 0.25)) (* (PI) 0.25)))

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)}} \]

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\log \color{blue}{\left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot f\right)} \cdot \frac{1}{4}\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot f\right)} \cdot \frac{1}{4}\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   5. lower-PI.f64 95.7

      \[\leadsto \frac{\log \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot f\right) \cdot 0.25\right)}{0.25 \cdot \mathsf{PI}\left(\right)} \]

7. Applied rewrites 95.7%

   \[\leadsto \frac{\log \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}{0.25 \cdot \mathsf{PI}\left(\right)} \]

8. Final simplification 95.7%

   \[\leadsto \frac{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}{\mathsf{PI}\left(\right) \cdot 0.25} \]
9. Add Preprocessing

Alternative 5: 95.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot \frac{4}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log (* (* (PI) 0.25) f)) (/ 4.0 (PI))))

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)}} \]

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\log \color{blue}{\left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot f\right)} \cdot \frac{1}{4}\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot f\right)} \cdot \frac{1}{4}\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

   5. lower-PI.f64 95.7

      \[\leadsto \frac{\log \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot f\right) \cdot 0.25\right)}{0.25 \cdot \mathsf{PI}\left(\right)} \]

7. Applied rewrites 95.7%

   \[\leadsto \frac{\log \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}{0.25 \cdot \mathsf{PI}\left(\right)} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right)}{\color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right)}{\frac{1}{4}}}{\mathsf{PI}\left(\right)}} \]

   4. div-inv N/A

      \[\leadsto \frac{\color{blue}{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right) \cdot \frac{1}{\frac{1}{4}}}}{\mathsf{PI}\left(\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{4}}{\mathsf{PI}\left(\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}} \]

   7. lift-/.f64 N/A

      \[\leadsto \log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right)} \]

   9. lower-*.f64 95.5

      \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)} \]

9. Applied rewrites 95.5%

   \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(f \cdot \left(\mathsf{PI}\left(\right) \cdot 0.25\right)\right)} \]

10. Final simplification 95.5%

    \[\leadsto \log \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot \frac{4}{\mathsf{PI}\left(\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ (PI) 4.0)) (log (/ (+ (exp (* (/ (PI) 4.0) f)) (exp (- (* (/ (PI) 4.0) f)))) (- (exp (* (/ (PI) 4.0) f)) (exp (- (* (/ (PI) 4.0) f)))))))))